The Duistermaat-Heckman formula and the cohomology of moduli spaces of polygons

Abstract

We give a presentation of the cohomology ring of spatial polygon spaces $M(r)$ with fixed side lengths $r \in \mathbb R^n_+$. These spaces can be described as the symplectic reduction of the Grassmaniann of 2-planes in $\mathbb C^n$ by the $U(1)^n$-action by multiplication, where $U(1)^n$ is the torus of diagonal matrices in the unitary group U(n). We prove that the first Chern classes of the $n$ line bundles associated with the fibration $r$-level set $\rightarrow M(r)$ generate the cohomology ring $H^* (M(r), \mathbb C).$ By applying the Duistermaat--Heckman Theorem, we then deduce the relations on these generators from the piece-wise polynomial function that describes the volume of $M(r).$ We also give an explicit description of the birational map between $M(r) $ and $M(r')$ when the lengths vectors $r$ and $r'$ are in different chambers of the moment polytope. This wall-crossing analysis is the key step to prove that the Chern classes above are generators of $H^*(M(r))$ (this is well-known when $M(r)$ is toric, and by wall-crossing we prove that it holds also when $M(r)$ is not toric).